Equation solving mechanism



' Jan. 5, 1954 EQUATION SOLVING MECHANISM Filed May 7, 1947 8 Sheets-Sheet l \J 58 94 57 I O f LL52 Q 6 u L xm J5 5g fi 5 Z5 35 21! 4 27 26 24 44 INVENTOR. Lewis I 1mm .197" T0 KNEYS Jan. 5, 1954 L. w. lMM

EQUATION SOLVING MECHANISM 8 Sheets-Sheet 2 Filed May 7, 194? INVENTOR.

lewl fi I? m flrraremiys Jan. 5, 1954 L. w. IMM 2,665,069

EQUATION SOLVING MECHANISM Filed May 7, 1947 8 Sheets-Sheet 3 INVENTOR. Lewis W I m m w fa K7 TTORNEvs Jan. 5, 1954 w MM 2,665,069

EQUATION SOLVING MECHANISM Filed May 7, 1947 s Sheets-Sheet 4 INVENT OR.

flTTOENEYJ Jan. 5, 1954 L. w. lMM 2,665,069

EQUATION SOLVING MECHANISM Filed May 7, 1947 8 Sheets-Sheet 5 y 1 I: ILZJIIII' Jan. 5, 1954 L. w. [MM 2,665,069

EQUATION SOLVING MECHANISM Filed May 7, 194'? 8 SheetsSheet 6 I I 4 I FlIEi l'7 HVVENTUR. Jew/'5 W //77/77 ATTORNEYJ L. W. [MM

EQUATION SOLVING MECHANISM Jan. 5, 1954 Filed May 7, 1947 8 Sheets-Sheet '7 BY/Z IN V EN TOR. [aw/j 14 //77/77 ATTORNEYLS Jan. 5, 1954 L. W. IMM

EQUATION SOLVING MECHANISM Filed May 7 1947 8 Sheets-Sheet 8 INVENTOR. [aw/'5 VIZ/r27? BY M/ ,4 TTQRN Y- Patented Jan. 5, 1954 EQUATION SOLVING MECHANISM Lewis W. Imm, Pacoima, Calif., assignor to Libraseope, Incorporated, Burbank, Calif., a corporation of California Application May 7, 1947, Serial No. 746,512

8 Claims.

The present invention relates to improvements in mechanisms for solving equations of the type in which the solutions, graphically represented, approach a limit value or values by progressively decreasing increments. Examples of such curves are parabolic curves, which approach a single limit value by such progressively decreasing increments; sinusoidal curves, which similarly approach two opposite limits; and curves which approach their limits as an asymptote, but theoretically do not ever quite reach such limits. Such mechanisms include, but are not limited to, devices for extracting exponential roots and devices for converting a function of an angle into an angular value, as well as devices for solving more complex equations.

Previously known mechanical devices for the solution of such equations by the transmission of proportional movements from input to output elements of the computer have been either inaccurate or inoperative in that part of their range in which the output value is approaching its limit, and have been wholly incapable of passing through a limit value to register a nega- 'tiVe output value. This is true because, as the output value approaches a limit value, the mechanical disadvantage encountered in driving the output registering device becomes progressively greater; theoretically becoming infinite when the output value coincides with the limit value. As a practical matter, however, the mechanical disadvantage becomes insurmountable somewhat before the limit value is reached, because of the friction inherent in any mechanical system.

It is the principal object or" the present invention to extend the range of calculation of mechanisms of the general class above described.

It is a further and more specific object of the present invention to make possible the extension of the range of calculation of such mecha- 'nisms to and beyond limit values, so that negative output values may be obtained.

The foregoing, together with additional objects and advantages which will be apparent, are attained by the application of a new principle according to which the output element of a computing mechanism of the type described, instead of being driven by mechanical power from the input element transmitted "through the computing mechanism, is driven by a separate source .of mechanical power, transmitted reversely through the computing mechanism and controlled by the input element so that the extent of the movement so imparted to ,the computing mechanism matches the extent of the movement which should, but cannot under the circumstances outlined above, be imparted to it by the input element.

The best mode in which the application of this principle has been contemplated is described in the following specification disclosing preferred embodiments of the invention adapted for the solution of various exemplary equations; reference being had to the accompanying drawings, in which:

Figure 1 is a front elevational view of the machine;

Figure 2 is a cross sectional view taken on the line 2-2 of Figure 3 and looking towards the rear of the machine;

Figure 3 is an elevational view looking from the left to the right of Figure 2 and at right angles thereto;

Figure 4 is an elevational view, partly in cross section, on the line 4-4 of Figure 5;

Figure 5 is an elevational view partly in cross section, showing the front plate of the machine and certain of the parts mounted thereon taken substantially on the line 5--5 of Figure 4;

Figure 6 is a view looking toward the rear from the rear plate to show certain of the parts mounted thereon;

Figure '7 is a perspective view looking toward the front wall of the machine and with the rear wall removed so that the parts may be more clearly seen;

Figure 8 is a plan view taken on substantially the line 8-8 of Figure 2;

Figure 9 is an exploded view of one of the differential mechanisms employed;

Figure 10 is a face view of the diiferential mechanism shown in Figure 9 and taken on substantially the line 10-10 of Figure 11;

Figure 11 is a cross sectional view taken on substantially the line H--ll of Figure 10;

Figure 12 is a similar cross sectional view taken on the line l2 l2 of Figure 10;

Figure 13 is a side elevational view partly in cross section taken on substantially the line l3l3 of Figure 12;

Figure 14 is a face view of the spider shown in Figure 13;

Figure 15 is a top plan view of the spider shown in Figure 14;

Figure 16 is an exploded view of a second type of differential mechanism;

Figure 17 is a circuit diagram of the control circuit of the motor l3;

Figure 18 is a diagrammatic illustration of a cube root extracting mechanism embodying the present invention;

Figure 19 is a diagrammatic illustration of a fourth root extracting mechanism embodying the present invention;

Figure 20 is a diagrammatic illustration of a triangle solving mechanism employing two matching arrangements according to the present invention to solve for an unknown angle and for the hypotenuse simultaneously;

Figure 21 is a detail view, in perspective, of a sine converter utilized in the arrangement of Figure 20; and

Figure 22 is a partly diagrammatic illustration of a linkage type triangle solving mechanism embodying the present invention.

First, the invention will be described as it is preferably embodied in a right triangle solving mechanism in which extraction of a square root is necessary to ascertain the length of the hypotenuse when the lengths of the two sides are known.

In the embodiment first described, values proportionate to the known lengths of the sides of the triangle are squared by function computing mechanisms of the form described and claimed in my copending application Serial Number 508,231, filed October 29, 1943, now Patent No. 2,485,200, granted October 18, 1949, although, as hereinafter explained, the present invention is not limited in its application to function computing mechanisms of that form. The outputs of the two function computing mechanisms are then combined by differential gearing to give an output equal to the sum of the squares of the lengths of the known sides and equal to the square of the length of the hypotenuse,

The length of the hypotenuse is then ascertained by extracting the square root of this value; but the power required to drive the square root extracting mechanism, instead of being furnished through the abov mentioned squaring m c n ms d difi re'ntial aring, is supp i d by a separate servomotor. This motor power is applied at the output, indicating or root side of the square root extracting mechanism. The motor thus turns th input or squared value shaft of the square root extracting mechanism the pow r being t smitted to t f m the output or root value shaft side. control of the servo-motor between the differential gearing output shaft and th input shaft of the square root extracting mechanism, the rotation of these shafts is matched, thus insuring that the servomotor moves the square root extracting mechanism to match the extent of movement which should be imparted to it by the difierentia1 gearing output shaft. Throughout this specification, unless specifically defined otherwise, the terms input and output have been used in a sense to indicate the data-transmitting direction of operations. This does not coincide in all parts of the structure with the direction in which power to operate the mechanism is being supplied. In the portion for extracting the value of c, for example, the driving power is inserted on the answer or output side, and fed back to drive the elements into matching position with the a +b elements. Thus the da a-tr nsmiss n sequ nce is from th and b values to a and 11 values, thence to a plus 13 Values (equal to 0 thence finally to the e value: while there are two poweretransmission so.- qiien e one f m a and b to a? a then to 11 -142 and the other from c to 0 Under hese conditions, th outp i the By disposing the D square root extracting mechanism will accurate- 1y represent the length of the hypotenuse, down to and including a zero value thereof; a range of computation impossible with previous devices because of the mechanical disadvantage encountered in attempting to drive any square root extracting mechanism down to any close approach to a zer root value.

Referring specifically to the accompanying drawings, illustrating this embodiment, the parts of the machine are on a front plate I and a rear plate 2 which are connected by spacing and tie members 3.

Considering that we have a right triangle having an altitude a, a base b and a hypotenuse c, the values of a and b may be entered into the machine by rotating the input knobs 4 and 5, respectively. The knob 4 is secured to an input shaft 6 to which is secured a dial i having calibrations adapted to be read relative to a reference point 8. Also secured to the shaft 5 is a gear 9, a pinion l0 and a cone Ii. The pinion It drives a gear 82 secured to a shaft E3 to which is also secured pinion M which drives a gear 15 secured to a shaft It to which is secured a dial I! having calibrations thereon adapted to be read relative to a reference point 18.

In like manner the input knob 5 is secured to a shaft l9 to which is secured a disk havin reference characters thereon adapted to be read relative to a reference point 2!. Also secured to the shaft I9 is a pinion 22, a gear 23 and a cone 24. The pinion 22 drives a gear 25 secured to a shaft 26 which carries a pinion 2'! which drives a gear 28 secured to a shaft 29, to which is secured a disk 39 having calibrations thereon adapted to be read relative to a reference point 3|.

The gear 9 drives a gear 32 rotatably mounted on a shaft 33. The gear 32 serves as one input for a differential mechanism to be presently described.

As fully explained in my copending application, Serial No. 508,231, previously referred to, now U. S. Patent No. 2,485,200, I have provided a cone, cylinder and differential mechanism whereby a number may be squared. In Figure 8 I have shown such a squaring mechanism which is the squaring mechanism to square the leg I) of the triangle. The cone 24 is provided with two helical threads and 35 each forming an Archimedean spiral and extending throughout the length of the cone. This cone does not extend to a point, and a typical example of the type of cone employed would be one having a small radius of .2, the radius of its large end being .8" so that its average radius is .5" so that the increase in radius per turn of the cone would be .015" if there were convolutions of each of the threads 34 and 35. Attached to the small end of the cone at the point 35. is a cable 3'? which is wound around the thread 3G and passes from the cone to a cylinder 38 on to which it is wound and its other end is attached to the point 39 of the said cylinder. A second cable 13 is secured to the cone at its large end at the point 41 and is wound around the thread 35 to where it passes from the cone to the said cylinder 38 to which it is attached at a point 42.

The gear 23 drives a gear 43 forming a part of a di ferential mechanism. The ear 43 loosely mounted on the shaft 44, on which the c l nder s m un ed, The di ferential mechan m ma be bett r nderst od. by referrin to Figures 9 to 15 inclusive. The cylinder 38 is secured to a 9111191; 45 and drives Pinions 46 mounted on arbors 4'! carried by arms 48 of a spider 49. The pinions 46 mesh with and drive pinions 50 mounted on arbors 51 carried by arms 52 of the spider. The pinions 50 mesh with a pinion 53 secured to the gear 43. An output gear 54 is secured to the spider 4G by any suitable means such as being pressed thereon. A cover 49a is secured to cylinder 38 to protect a portion of this gearing.

If the cone tapered to a zero point the differential mechanism would not be necessary, but it is not practical to use a cone tapered to a zero point in the machine. The net result of this differential mechanism is to transfer the zero point of the theoretical cone to a given point on the actual cone such as the small end thereof. As fully explained in my said copending application, this is accomplished by providing the differential mechanism and provided the diameter of the zero point selected for the cone,

such as its small end, is to the diameter of the cylinder as the ratio between the gears 23 and 43. This may be further explained as follows:

If the cone had a small radius of .2", a, large radius of .8 and if the radius of the cylinder is .5" and if the pinion 23 has a8 teeth and the gear 43 has 120 teeth so that the small radius of the cone is to the radius of the cylinder as 1 is to 2 and the gear ratio between the gears 23 and 43 is as l is to 2 /2, in radius per turn of the cone is .015, and if the output gear 54 has 96 teeth and drives a gear 55 of 48 teeth, the following computation is correct:

Y No. of a N o. of I\ o. of

No. of turns of iggi revel. of l revel. subggg' g cone comb; dia. cyltracted by 55 I inder 38 differential Shag; 415 415 4 O15 445 860 8 i 069 475 1.335 1. 2 135 505 1. 840 1.6 240 535 2. 375 2. O .375 565 2 940 2. 4 540 595 3 535 2. 8 735 625 4.160 3. 2 i 960 555 4. 315 3. 6 1. 215 685 a 500 4. 1. 500

It will be noted from the above tabulations that the shaft 56 is rotated proportional to the square of the number of rotations of the cone. The gear 62 is secured to the shaft 56 and drives a differential gear 63.

The squaring mechanism for the input for the leg a may be identical to that described above for the input value b. However, there may be cases in which the leg arepresenting the altitude of the triangle may be considerably shorter than the leg b representing the base. Suppose, for instance, the limit of the altitude should be found never to exceed /2 of the length of the base. We might then employ the entire cone 1! to enter only /2 as much value as the cone 24, and yet We desire to use the entire cone H to enter this maximum value 28. This could be done by providing a suitable reduction gearmg.

The cone H drives a cylinder in precisely the same manner as the cone 24 drives the cylinder 38. There is also provided a differential mechanism identical with that heretofore described and interposed between the cylinder 57 and the gear 32 which drives an output gear 58 for the differential. This gear 53 drives a gear 58 secured to a shaft 613 to which is secured a pinion 6i which drives a, differential gear 64.

and if the increase It is therefore apparent that the diiferential gear 63 is driven an amount proportional to the square of the value entered for the leg I) while the differential gear 64 is driven an amount proportional to the square of the value entered for the leg a.

The difierential mechanism shown in Figure 16 adds (L -Hi The gear 63 has secured thereto a pinion 65 while the gear 64 has secured thereto a pinion 66. The pinion I meshes with and drives pinions 88 carried by the difierential spider 10 which is secured to the output shaft 69. The pinion meshes with and drives pinions 61 carried by the differential spider it. Each of the pinions $7 meshes with its res ective pinions E58 but they are slightly offset so that the pinions 67 do not mesh with the pinion 65 but are driven only by the pinion while the pinions 68 do not mesh with the pinion 58 but are driven by the pinion 65. It is therefore Obvious that the output shaft 89 will be driven an amount proportional to a +b The shaft 9 has secured thereto an arm H having a contact 12 mounted thereon. The arm H will therefore move an amount proportional to a +b The calculating mechanism for computing the value of the hypotenuse c from the square of that value is identical in construction to that heretofore described in connection with the col-- culation of the values for a and b and squaring the same, but is operated to find the square root instead of the square of an input value, and the driving power is fed through the mechanism from the solution or output side instead of from the data or input side. A servomotor l3 drives a gear 14 mounted on the rear wall of the machine on a shaft 15 to which is attached a cone 15, a pinion 11, a gear 78 and a disk 59 having calibrations thereon adapted to be read relative to a reference point 80. The pinion ll drives a gear 8| secured to a shaft 82 to which is attached a pinion 83 which drives a gear 84 secured to a shaft 85, to which is attached a dial 86 adapted to be read relative to a reference point 81.

The cone 16 is identical to the cones heretofore described and drives a cylinder 8% in identically the same manner as heretofore described in connection with the cylinders 38 and The gear 18 drives a gear 89 which performs exactly the same function as the gear 43. A differential mechanism identical to that shown in Figure 9 is interposed between the cylinder 58 and the gear 89 and has a gear 9!! identical to the output gear 54. This gear 90 drives a gear 9! secured to a shaft 92 which extends through to the rear wall of the machine and has secured thereto a pinion 93 which drives a gear t l, which is coaxial with the shaft 69 and is rotatably mounted thereon. Secured to the gear E i is an arm 95 and a contact 86 near the outer end thereof. It is obvious that the gear iii. and the arm 95 are rotated an amount corresponding to 0 The arms H and 95 are initially set so that when a +b =C the said contacts '12 and 96 will touch each other.

The mm H is insulated from the shaft 59 but the arm 95 is not insulated. An arm 3? is secured to a stud 98 and carries an arm 99 insulated from the arm 91 by insulation Hill. An electrical lead It]! extends to the arm as which bears against the insulated arm H near its cen ter point. Therefore, when the contacts 72 and 95 are closed, the circuit is completed from the lead Hli through the arm 99, arm "5!, contacts 12 and 96 and arm 95, to the framework of the machine to which is attached a return lead )2 thereby completing a circuit. The leads I9! and I02 control the servo-motor so that when the circuit is closed by the contacts 12 and 96 touching each other, the motor operates in one direc tion, and when they are separated the motor operates in the opposite direction. Many such motors and motor controls are well known; an example of a standard and well known type being diagrammatically illustrated together with its circuit form no part of the present invention,

and any reversible motor similarly controlled through a reversing relay or otherwise can be as readily utilized.

The provision of a reversible servomotor of either the type just described or the type described in connection with the embodiment shown in Figure 22 makes possible the continuous indication of a result during changes of any input value or values, regardless of whether such input value or values is increasing or decreasing. b

The net result is that when the motor has rotated the arm 9%: an amount equal to the rotation of the arm II, the circuit is closed and the indicators 79 and 86 now represent the value of c for the reason that in the right triangle (2 equals a +b and the arm H has been rotated in an amount to represent a +b and the arm 95 has been rotated in the amount to represent 0 As soon, however, as the contacts 72 and 96 are closed, the motor is reversed with the result that the arm 85 starts to move so as to break the said contacts 72 and 96 but instantly the motor is now reversed again bringing the contacts together again. This back and forth movement of the arm 95 is exceedingly small and in fact, so small as to be negligible, so that we can consider that the indicators representing the 0 value remain stationary unless the value of a or 22 should be changed. If either of them were changed so as to move the arm II clockwise as viewed in Figure I 6, the arm 95 would follow it and if the arm H were moved anti-clockwise the arm as would back away from itnot by the arm ll pushing the arm 95, but because through the control mechanism the arm 95 is constantly being moved slightly away from and then towards the arm i I.

We have heretofore considered that the two legs a and b were known, and the hypotenuse c was to be determined from the equation a +b =c If the hypotenuse and one leg are known, and if it is desired to calculate the value of the other leg, we can call the input knob i, the hypotenuse knob, and the input knob 5 the knob of the known leg. The expression would now become c b =a or c a =b The only structural change required would be to place an idler gear between the pinion SI and the differential gear B l (Figure '7) so that the differential mechanism 53, 6d would subtract instead of add. The indicator l8 would now show the length of the third side.

The machine, however, can be utilized without any structural changes to obtain the value of an unknown leg when the hypotenuse and one leg are known. This can be done by entering the value of the known leg through either the input knob d or the input knob 5. The other knob can then be rotated so as to bring the indicators I9 and 3t to the value of the hypotenuse. The dials associated with this last named input knob will give you a direct reading of the unknown leg. In Figure 18 of the drawing, there is diagrammatically illustrated the manner in which the invention is embodied in mechanism for extracting the cube root of an input quantity. In this embodiment, the input shaft I if is rotated an amount proportionate to the value of the quantity of the cube root of which is to be extracted. This acts to separate a pair of contacts HI, H2 corresponding to the contacts 52 and 86 of the triangle solving mechanism just described; the contact III being secured to the shaft IIil so as to be moved away from the contact H2 by rotation of the shaft. The contacts III, II2 control the circuit H3, I I of a servomotor Us the shaft i It of which drives a cone and cylinder squaring mechanism II? which includes the cone Illa, the cylinder lb, and the differential mechanism I lie. The output shaft I I8 of this squaring mechanism Ill will therefore be rotated in amount proportionate to the square of the quantity by which the input shaft iii: is rotated by the motor H5.

The shaft H6 of motor H5 also acts through shaft M9 to drive one input leg of each of two differentials tilt] and I2I, a second input leg of which differentials being driven by the output shaft N8 of the squaring mechanism iii. The inputs of the differentials i2i3 52! are arranged so that the diiferential I29 adds to the output of the squaring mechanism I it a quantity equal to the input fed thereinto by the shaft H9, while the differential I2! subtracts from the output of the squaring mechanism II? a quantity equal to the input transmitted by shaft I I9.

The output of the differential $25 is fed by shaft I22 through a second squaring mechanism I23 of the same general type as the squaring mechanism Ill, and the output of the squaring mechanism I23 is fed by shaft I25 into one input leg of a differential I25. Similarly, the output of the differential IEI is fed by a shaft F25 through a third squaring mechanism 52? of the same type as the squaring mechanism I5! and the output of the squaring mechanism I22 is fed by shaft I28 into another input leg of the differential I25. The inputs from shafts I2 3 and I28 into the differential I25 are directionally related so that the output of the squaring mechanism I23 is subtracted from the output of the squaring mechanism [21, the result being that the output of the differential H5 is equal to the cube of the quantity represented by the rotation of the shaft H6 of motor H5. The output of differential I25 rotates a shaft i35 which carries the other of the two previously mentioned contacts i I l, I I2 acting to move the contact H2 into engagement with the contact Iii, which as previously mentioned was positioned in accordance with the input quantity of which the cube root was desired. When contact H2 moves into contact with the complementary contact II E, the motor H5 is arrested just as in the case of the previously described triangle solving mechanism, and at this stage it will be observed that the shaft I I 6 of the motor I15 has been rotated an amount proportionate to the cube root of the quantity represented by the rotation of the input shaft iIiI. The result of the computation may be conveniently indicated by an indicating device I3l attached to an opposite end of the shaft II6.

In Figure 19 of the accompanying drawing, there is diagrammatically illustrated an embodiment of the present invention adapted for the extraction of the fourth root of an input quantity.

This arrangement is generally similar to that illustrated in Figure 18, although mechanically somewhat simpler; the input quantity the fourth root of which is desired being entered by rotation of the shaft I35 an amount proportionate to the value of the quantity. By separation of the contact I36 from the contact I31, the rotation of the input shaft I35 causes rotation of the shaft of the servomotor I40, the shaft I4I of which drives a squaring mechanism I42 so that the output shaft I43 of the squaring mechanism is rotated an amount proportionate to the square of the quantity represented by the rotation of the shaft MI. The ouput shaft I43 of the squaring mechanism I42 in turn drives the input of a second squaring mechanism I44 the output shaft of which I45 is thus rotated an amount proportionate to the square of the quantity represented by the rotation of the output shaft I43 of the first squaring mechanism I42 and thus by the fourth power of the quantity represented by the rotation of the shaft I4I of the motor I40. The contact I3! is moved by the shaft I45 and brought into engagement with the shaft I36 when the shaft M! of the motor I40 has rotated an amount equal to the fourth root of the quantity represented by the rotation of the input shaft I35, and such a fourth root may be conveniently indicated by an indicating device I45 rotated by the opposite end of the shaft I4I.

In Figure 20 there is illustrated diagrammatically an arrangement whereby the present invention is utilized by employing a plurality of servo-motor balancing arrangements to simultaneously solve a series of equations, the second equation in the series utilizing as an input quantity the quantity found by the solution of the first equation in the series.

The first equation solved by the arrangement illustrated in Figure 20 is that solved by the triangle solving mechanism first described in this specification, i. e., c= /a +b The arrangement for solving this equation is diagrammatically illustrated in this figure of the drawing with the same reference numerals applied to the elements as were applied to the parts shown in full in Figures 1 to 16, inclusive, of the accompanying drawing. The input corresponding to the altitude of the triangle is entered by rotation of the shaft 6 by means of knob 4, while the input corresponding to the base of the triangle is entered by rotation of shaft I9 by knob 5. These quantities are squared by the cone and cylinder squaring mechanism comprising cone II and cylinder in the case of the first input and cone 24 and cylinder 38 in the case of the second input. The outputs of the differentials associated with the respective squaring mechanisms are transmitted by gears 58 and 54, respectively, to a differential the inputs of which are gears 54 and 63, respectively, so that the squares are added together and the sum of the squares is fed out through the output shaft 63 which adjusts the electrical contact I2. Separation of the contact I2 from the contact 96 effected by displacement of the output shaft 69 in this manner causes rotation of the shaft of the servo-motor I3 in an amount proportionate to the square root of the sum of the squares of the altitude and the base of the triangle: tne rotation of the shaft of the servo-motor, which feeds in power from the root side, being squared by the cone and cylinder mechanism comprising the cone I6 and the cylinder 88 and the c-squared values fed back by gear to the electrical contact 36 for comparison with the a +b values, all as previously described.

In order to simultaneously solve for the value of the angle included between the hypotenuse and the base of the triangle, a contact I50 is simultaneously displaced by the rotation of the altitude setting knob 4 transmitted through shaft do as shown in Figure 20. Separation of the contact I50 from its cooperating contact I5I causes operation of a second servo-motor I52 rotating the shaft I53 of the motor I52 by an amount which, as hereinafter explained, will be proportionate to the angular value of the angle included between the hypotenuse of the triangle and the base thereof, which angular value may be conveniently indicated on an indicator I54.

The angular value represented by the rotation of shaft I53 is converted into a value proportionate to the sine of the angle by mechanism shown in detail in Figure 21, which comprises a crank arm I55 secured to the shaft I53 and having rotatably mounted thereon at I56 a gear I57 meshing with a fixed ring gear I58. Fixed to the gear I51 is a plate I59 having pivotably connected thereto at I60 an output rod I6I, the arrangement being such that the output rod I6I will have imparted to it an axial movement proportionate to the value of the sine of the angle represented by the rotation of the shaft I53.

The axial movement of the rod I6I which is proportionate to the sine of the angle included between the hypotenuse and the base of the triangle is converted into proportionate rotational movement and fed additively, as an input, into one input leg of a differential I65 on the one hand, and subtractively as an input into one input leg of a second differential I66. Simultaneously both the difierential I65 and the differential I56 receive as second inputs a movement proportionate to the value of the hypotenuse of the triangle, this movement being fed from the output shaft of the servo-motor I3 by shafts 15a and 15b. The output I6! of the differential I65 is thus proportionate to the valueof the hypotenuse plus the sine of the angle included between the hypotenuse and the base while the output I68 of the differential I66 is proportionate to the value of the hypotenuse minus the sine of the angle included between the hypotenuse and the base.

The output value represented by the rotation of the shaft I67 is squared by a cone and cylinder squaring mechanism ill and transmitted by the output shaft HI of said squaring mechanism to a differential Iit, while the output value represented by the rotation of the shaft IE8 is squared by a squaring mechanism II3 the output of which transmitted by shaft I14 as a second input to the differential I72. The relation of the inputs III and N4 of the differential I52 is such that the quantities represented by the two inputs are subtracted from each other, and the out put H5 of the differential H2 is thus proportionate to the value of the hypotenuse of the triangle multiplied by the sine of the angle included between the hypotenuse and the base thereof; being thus proportional also to the value of the altitude of the triangle.

The previously mentioned contact I5I is positioned in accordance with the-angular position of the out-put shaft N of the differential H2, andthus when the value represented by the output shaft I75 is equal to the value of the altitude of the triangle in accordance with which the companion contact i53 is positioned by the shaft 6a, movement of the shaft !53 by the servo-motor I52 will cease and under these circumstances the angular positioning of the shaft I53 will be proportionate to the value of the angle included between the hypotenuse of the triangle and its base.

The arrangement thus provides a convenient means for simultaneously solving for the hypotenuse of a right triangle and the value of the angle included between the hypotenuse and the base given the value of the base and the altitude of the triangle.

Although the present invention has been described in the preceding embodiments as applied to computing mechanisms utilizing cone and cylinder squaring mechanisms, it is of general utility in connection with other types of computing mechanisms, as illustrated in Figure 22 of the accompanying drawings in which the invention is shown applied to a triangle solving mechanism employing a conventional linkage type of multiplying mechanism.

In the embodiment of the invention illustrated in Figure 22, means are provided for determining the length of the hypotenuse of a right triangle when the values of the base and the altitude are known. For mechanical convenience, the conventional mathematical formula is transposed into the form a =(e+b) (c-b).

Here 0 is the hypotenuse; a is the altitude; and l) is the base of the triangle.

Referring specifically to Figure 22, the value equal to the length of the base of the right triangle to be solved is fed into the instrument by rotation of a shaft 236', which may for convenfence have hand knob 26! attached thereto, although if the input value is to be entered other" wise than manually, as from another computing mechanism or distance determining mechanism, the manual setting knob 231, may, of course, be omitted. Preferably, but not essentially, this value is, also indicated on a dial 2E2 connected to .the'shaft 28s by gears. 263, 26 i, 2&5, 2G3. and 23?,

said dial being movable with respect to fixed indi cia 268. on a. frame 252 in which all of the parts herein described are mounted.

The value equal to the altitude of the triangle s fed into the instrument by rotation of a shaft 21 t which likewise may have attached thereto for convenience a hand knob 2l2 and similarly be connected to a dial 2E3 by gears 2E4, 2i5, and 216, said dial being movable with respect to indicia 2|.l on the frame 2 l0 and being calibrated, as, shown, so that the actual rotation of the shaft ill will be proportionate to the square of the value representing the altitude of the triangle,

While the Values inscribed on the dial 2L3 will.

give a direct reading or the altitude value itself.

The computation of the valve (0+b) and the value (cb) is accomplished by two conventional difierential gear mechanisms as and 22f, the previously mentioned gear 282 constituting an input gear of the differential 22% whereby the base value is entered negatively into that differential mechanism. The gear 2M1 is connected by an idler. ear 222 with aninput gear 22.3; of the secae'cisbee l2 and differential mechanism 22| so that the base value is simultaneously entered positively into the differential 22E.

The value or the hypotenuse of the triangle is determined by matching against the square of the value of the altitude, entered as previously described by rotation of shaft 2! i, the product of the multiplication (c+b) (0-17) and for this purpose a servo-*notor is utilized to feed into both of the differentials 22!] and 221 a positive quantity equal to the calculated value of the hypotenuse. For this purpose the servo-motor 225 is connected by a gear 226 with a second input gear 221 of the differential 22! which gear 22'! meshes directly with a second input gear 228 of the difierential 220.

The servo-motor 225 is controlled by a conventional system comprising a generator 232 directly actuated by rotation of the altitude input shaft 2!! and a control transformer 253i actuated by the product registering element of the multiplying mechanism as hereinafter described, the arrangement being such that rotation of the altitude input shaft'ZH by displacing the control element of the generator 23f; causes the servo-motor 225 to commence operation, which operation continues until the displaceable element of the control transformer 23! has been displaced by the product registering element of the multiplying mechanism an amount equal to the initial displacement of the altitude input shaft 2 i l. Servomotor controls of this general character are well known in the art and therefore are not described in detail herein.

Operation of the servo-motor 225 initiated by displacement of the altitude input shaft 2| I, as above described, therefore causes rotation of the output gear 235 of the differential 222 by an amount equal to (kg-21 and of the output gear 236 of the differential 22! by an amount equal to (c-i-b). These output values of the differentials 22d and 22! are then multiplied bya, conventional linkage form of multiplying mechanism to give a product equal to the value of a as set forth in the above formula.

The multiplying linkage embodied in the instrument disclosed in Figure 22 is of the general type disclosed in Imm Patent No. 2,394,180, dated February 5, 1946, being particularly described at page 4, column 2, beginning at line 62 therein. In the instrument shown in Figure 22, this mechanism is illustratedas comprising an element 245 pivotal'ly mounted on the-frame 2 i c at 24 I, a mov able link 2&2 pivotally connected to the element 240 at 2. 33, asecond link 2% having a pivotalconnection with the link 2 32 at 2 35, and a prod uct. registering element 255 pivotaliy mounted on the frame at 2 and having a pivotal connec tion with the link 26 5 at 2&8; the spacings be tween. the pivotal connections 2% and 2:33, 243 and 245, and 2&5 and 2438, all being equal.

The element 2th of this multiplier linkage is pivotally connected at 250 to a rod 25! actuated by a. rack Z552 meshing with a pinion 253 mounted on a. shaft 25 on which there is also mounted a pinion 255. meshing with the output gear 235 of the differential 22 8 this arrangement being such that the element of the multiplier linkage will be displaced an amount proportional to the rotation of the output of the differential 22c which is equal to the value (cb).

The pivotal connection 245 of the multiplier linkage just described is connected to a rod 268- multiplier linkage 2.12 245- and pivotally connected at its opposite end at 262 to a lever 263 which in turn is pivotally connected at 264 to the rod 26! The rod 253 is in turn connected to a rack 265 meshing with the pinion 266 mounted on the shaft 26? on which there is also mounted a pinion 258 meshing with the output gear 235 of the differential 225, this arrangement being such that the pivotal connection 245 of the multiplier linkage is displaced by an amount proportionate to the rotation of the output gear 236 of the differential 22! or by an amount equal to (c+b).

The structure of the multiplying linkage above described is such that, as particularly described in the above mentioned Imm Patent No. 2,394,180, the product registering element 246 will be displaced about its pivot 2&1 by an amount equal to the product of the displacement of the pivotal connections 2 13 and 245, and the movement of the product registering element 246 will thus be equal to the product of the expression (+2?) (c-b) or equal to the square of the value of the altitude of the triangl Displacement of the product registering element 2% is therefore used to control the stopping of the operation of the servo-motor 225 initiated by displacement of the displaceable element oi the generator 23c by rotation of the shaft 2 H as an incident to the entry of the altitude value. The connection between the multiplier linkage product registering element 266 and the control transformer 23| which accomplishes this purpose comprises a link 210 pivotally connected to the output element 246 at 21! and at its lower end pivotally connected at 2'i2 to a lever 213 pivotally mounted on the frame of the instrument at The lever 213 is pivotally connected at it opposite end at 215 to a rod 216 having at its lower end a rack 21'! meshing with the pinion 2'58 mounted on a shaft 219 upon which there is also mounted a pinion 280 meshing with a further pinion 28! mounted on a shaft 282 upon which there is also mounted a pinion 233 meshing with a pinion 284 mounted on the shaft 285 or the control transformer 23! this arrangement being such that when the shaft 285 is rotated by the arm 24B of the multiplier linkage, an amount equal to the initial rotation of the shaft 2H by the input handle 212, operation of the servo-motor will be arrested, since at this stage of the operation of the servo-motor a hypotenuse value has been entered which will balance the formula a :(c+b) (cb). The arresting of the servo-motor thus takes place at a point when the rotation of the servo-motor shaft has correctly registered the value of the hypotenuse, and this value may be conveniently registered or indicated on a scale 290 connected to the servo-motor shaft by gears 29!, 292, 293, 221 and 226, said scale being movable with respect to indicia 295 carried by the frame 210.

It is particularly evident in contemplating this embodiment of the invention that ii an attempt were made to displace the pivotal connection 245 of the multiplier linkage by the application of force to the pivotal connections 263 and 248, respectively, and thus to register the value of the hypotenuse by a displacement of the pivotal connection 2 eilected in such a manner, the force required to be exerted would increase to infinity as the pivotal connection 245 approached coaxial alignment with the pivotal mounting 24!. It would thus be a mechanical impossibility to move the pivotal connection 245 into axial alignment with the pivotal mounting 2M in this manner. However, according to the present invention, the

movement of the pivotal connection 245 into alignment with th pivotal mounting 24! to register a zero value for the hypotenuse of the triangle being accomplished by the servo-motor acting through gears 226, differential 22!, gears 2G3 and 256, rack 265, lever 263, and link 26! is accomplished with the utmost facility.

The same considerations apply in the other embodiments of the invention herein described, and it is thus seen that the present invention fully accomplishes the objects hereinabove set forth.

The servo-motor and controls therefor illustrated and described in connection with Figure l"? of the drawings, and the servo-motor and controls therefor illustrated and described in connection with l igure 22 of the drawings are merely examples of well known types of such mechanisms, and either type can be used in place of the other in a manner which will be obvious to those familiar with such devices.

Reference is to the article entitled Selsyn Design and Application by T. C. Johnson, published in Electrical Engineering Transactions for October, 1945, Volume 64, pages 703 to 708 inclusive for descriptions of a variety of mechanisms of the type in question including one of "he type referred to in connection with Figure 22 of the present specification, and it will be evident that this generator-and-transformer control could be substituted for the contact control of Figure 17 merely by utilizing the means for displacing the contact T2 to displace the generator,

utilizing the means for displacing the conto displace the transformer.

servounotor l3 of Figure 17 has windings which are unbalanced, so that the shunting of one of the two condensers by closure of the contacts causes reversal of the motor, while the opening of the contacts causes it to run in its normal direction. It will be evident that this type of motor and control could be substituted for the generator-and-transformer type of motor and control merely by utilizing the means for displacing the generator to displace the contact '12, and utilizing the means for displacing the transformer to displace the contact 96.

This application is a continuation-in-part of my application, Serial No. 508,233, filed October 1943, for a Triangle Solving Mechanism.

What I claim is:

1. In a computing mechanism of the class described, the combination of a first member adapted to be differentially adjusted over a range of positions representative of a series of values which are to be computed, output means differentially adjustable over a range oi positions representative of a series of values, power-operated driving means for effecting differential achustr nt of said output means, proportional mover. nt transmitting mechanism including a first intermediate element and constituting a mechanical connection between said driving means and said first intermediat element, adding means controlled by said driving means and said first intermediate element for registering the sum of the movements thereof, subtracting means controlled by said driving means and said first intermediate element for registering the difierence of the movements thereof, a second proportional movement transmitting mechanism including a second intermediate element and constituting a mechanical connection between said adding and constituting a mechanical :wmection '-between subtracting .-means.and said ,third intermediate element, .an (additional sub,tract ing .neans controlled by said second and =thir,d intermediate elements for registering the difference of the movements thereof, and :means jointly :controlled by said first member and said additional subtracting means for initiatingoperation of said driving means whenever the value represented by said first member is unequal togt-he value registered by said additional subtracting means and arr sting operation of said :driving means when values are equal.

2. In a computing device for effecting the simultanecus solution of ,apluralityof related equations, th combination of ;a firstaenuation :solvcomputing mechanism comprising. an :arljustable factor entering device, adjustable V-output means, power driving means for adjusting :said output means, and means jointly responsive to adjustment of said output means ,by saidgdriving means and to adjustment of said factor entering device for initiating and arresting adjustment of output means ;by.,said driving means; :asecnd equation solvingscomputing mechanism comp a second adjustableiactorentering jdevice, n ns controlled by said power driving means i'or adjusting said second adjustable factor ontering device, a secondadjustable output means, asecond power-driving means foradjusting said second output means, and means jointly responsite to adjustment or" said second output means by said second driving :means; to adjustment, of said second factor entering device by said first mentioned power driving means; and to adjustment of said jfirst mentioned factor entering device for initiating and .arrestingadjustment or" saidsecond output meansiby said second driving means.

.3. .Ina computer, the combination of a differentially adjustable value entering device, a servo-motor control including a pair of differentiallly adjustable, value-representing, follow-up elements, an operating connection between said value entering device and one of said adjustable elements, mechanical movement transmitting computing meanshaving an input member and an output member; said computing-means being incapable of being driven throughout-its entire range by power applied through said input memher, an operating-connection between-said input member and the other of said adjustable elements, a S6IMOelIlOl3OI,:a1'ld operatingconnection between said servo-motor and said output. memher, through which :output membersaid computing means may be driven throughoutits entire range, means jointly controlled ,bysaid adjustable elements for initiating operation of said servo motor whenever the-values representedibi' the respective positions of said follow-up elements are unequal and. arresting such operation when said values are equal, and result registering means driven by said servo-motor.

a. A computer according to claim 3 in which mechanical-movement-transmitting computing means comprises a multiplying mechanism,

5. A'ccmputer according to claim 3 in which said mechanical-movement-transmitting computing means comprisesxa :cone andcylinder having flexible driving. meansvinterconnectingithem.

16 In an equ tion solving computer, the come bination of 211E} lity ol differentially adjustable value entering -,nc,vices each adapted to be ,difierentia'lly adjusted to represent a known factor of an ier uation, a Sl"VO-:lIlOtOl control including a pair of differentiallyadjustable, value-representing, follow-up elements, a computin operating connection between said plurality of value entering devices and one of said adjustable elements, mechanical-movement-transmitting computing means having an input member and an output member; said computing means being incapable of being driven throughout its entire range by power applied through said input member, an operating connection between said input member and the other of said adjustable elements, a SfiEYOwi'l'lQliOl, an operating connection between said servo-motor and said output member, through which output member said computing means may be driven throughout its entirerange, means jointly controlled by said adjustable elements for initiating operationof said servo-motor whenever .thevalues represented by'the respective positions of said follow-up elements are unequal and arresting such operation when said values are equal, and result registering means driven bysa-id servo-motor.

.7..;In;a;computer for solving a right triangle,

represent the valueoi one known side or" a right triangle, iineans controlled .by .said device for squaring :a value entered thereby, aseconcl, differentially adjustable, value entering device adapteo to bediiferentially adjusted to represent thevalue of another known side or" a right tri angle, ineanscontrolledibysaid second device for soguaringla :value entered thereby, adding means controlled byisaid two squaring means and including an output element dirierentially positione-slzinaccordance with the sum of the squares of thevalues entered by said value entering devices, a servo-motor control including a pair of differentially adjustable, value-representing, follow-up elements, an operating connection between saidlcutputelernent and one of said followup elements, mechanicalemovement-transrnitting squarerootextracting means having an input memberand, an output member; said extracting means being incapable of being driven throughout itsentire rangeby power applied through said input member, an operatin connection between said input member theother of said follow-up elements, a -servc-1notor,;an operating connection bElEVQCZI SEK/id gserro-imotcr and said output member throughuvhi h output member, said extracting meansmay .be driven throughout its entire range, means jointly -.controlled -by said adjustable elements for initiating operation of said servo-rnot0r whenever the values represented by the respective positions of said follow-up elementsare-unequal and arresting such operation whensaid values are equal, and result registering means driven by said servoemotor.

"8. A computerfor solving. a right triangle according .to claim 7 in which said mechanical- LEWIS N. IMM.

ititefierences :on'following page) 17 References Cited in the file of this patent Number UNITED STATES PATENTS Number Name Date 2:472129 1,292,085 schantz Jan. 21, 1919 5 2,5Q5:258 1,529,817 Turner Mar. 17, 1925 2,040,918 Bristol May 19, 1 36 2,194,477 Maxson et a1 Mar. 26, 1940 number 2,349,118 Simpson May 16,1944 284 804 2,401,447 Wipff June 4, 1946 m Name Date Darlington Mar. 23, 1948 Gilbert Feb. 1, 1949 Streeter June 7, 1949 Robinette Apr. 25, 1950 FOREIGN PATENTS Country Date Great Britain Feb. 9, 1928 

